Maths F4....10

12.
(a)
Let α and β be the roots of the equation 2x + 5x + c = 0, where α > β.

Sum of the roots: α + β = -5/2 ...... [1]
Given that: α - β = 19/2 ...... [2]

[1] + [2]:
2α = 7
α = 7/2

[1] - [2]:
2β = -12
β = -6

Hence, the roots of the equation are 7/2 and -6.

(b)
Product of the roots:
(7/2) * (-6) = c/2
c = -42


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13.
(a)
Let α and 1/α be the roots of the equation -3x² + (7 - k)x + k = 0

Product of the roots:
α * (1/α) = k/(-3)
k/(-3) = 1
k = -3

(b)
The equation:
-3x² + [7 - (-3)]x + (-3) = 0
-3x² + 10x - 3 = 0
3x² - 10x + 3 = 0
(3x - 1)(x - 3) = 0
x = 1/3 or x = 3

Hence, the two roots are 1/3 and 3.